The mount is happiest with 12-14v DC at about 2 amps, it will draw less than 1/2 amp when just tracking. Good power and good power connections usually alleviate (or prevent) the majority of issues.

A set of anti vibration pads under the tripod often helps a bit, as does hanging some extra weight (10-20lbs) from the spreader bar.

If your using the mount in the cold (below freezing) you may also have to keep the hand controller warm otherwise it gets sluggish and hard to use. There are a number of ways to do this, wrapping a chemical hand warmer to the back is easy, a more robust solution is to also add a transparent cover (and still use the hand warmer).

Would a battery of 12V at 7A be sufficent? What regulator (AC to DC) would I need to get to connect to the mains power? I was told that the same regulator could be used to charge the battery, is this correct?

No, don't use a battery charger to power the scope, they aren't regulated (can often put quite a bit more than 14v into the scope) and usually can't supply the required amperage. You want a good quality 12-14v regulated DC power supply rated for at least 2 amps (you can get by with 1.5amps but 2 will give you some head room). Look for regulated 12v power supplies (which will often also be listed as 13.8v power supplies). You'll also need the correct power cable. The mount needs a 2.1 x 5.5mm plug. If you've got the Celestron power cable (it has a cigarette lighter style connector at one end) you're probably all set.

As I mentioned in my previous posting the mount will draw less than .5amp when tracking. This means with a 7amp/hr battery you could run the mount for 6-7hrs. First you usually don't want to take a battery down below 50% capacity very often. As a battery goes down it will also drop the output voltage a bit. The CG-5 goto electronics start to act "strange" below 10.5 volts. So a 7amp/hr battery is really a 3.5amp/hr unit for the purposes of planning. Just tracking at less than .5amp means that battery will last over 7 hours. You will likely need to slew (gotos) to various objects which will draw a bit more power so I suggested that this would mean the battery would give you between 6 and 7 hours of use.

Would a battery of 12V at 7A be sufficent? What regulator (AC to DC) would I need to get to connect to the mains power? I was told that the same regulator could be used to charge the battery, is this correct?

For brief observing sessions with few/short go-tos, yes. For more that that--all evening with fairly frequent slews, 17ah is the minimum. Trot down to Walmart (or Asda in the UK) and get a jump start battery pack.

And, as you've been told, don't try to run it on the charger. Unhappiness will result.

Yes the 2.1 x 5mm should be fine, there is usually a flat spring inside the socket side that presses against the outer barrel. The barrel of the plug may be a bit loose going in but should seat solidly when pressed all the way in.